Circle PYQs Set 6 (PDF)

Set 6 of 6 · Questions 151-164 of 164

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Circle · Set 6 (Q151-164)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
151. $Two tangents are drawn from a point P to the circle x^{2} + y^{2} - 2 x - 4 y + 4 = 0, such that the angle betweenthese tangents is tan^{- 1} (\frac{12}{5}), where tan^{- 1} (\frac{12}{5}) \in (0, π) If the centre of the circle is denoted by C and thesetangents touch the circle at points A and B, then the ratio of the areas of Δ PAB and Δ CAB is :$ (1) $11 : 4$ (2) $9 : 4$ (3) $3 : 1$ (4) $2 : 1$ 152. $Let circle C be the image of x^{2} + y^{2} - 2 x + 4 y - 4 = 0 in the line 2 x - 3 y + 5 = 0 and A be the point on C such that OA is parallel to x -axis and A lies on the right hand side of the centre O of C . If B (α, β), with β < 4, lies on C such that the length of the are AB is (1/6)^{th} of the perimeter of C, then β - 3 α is equal to$ (1) $3 + 3$ (2) $4$ (3) $4 - 3$ (4) $3$ 153. $Let the circles C_{1} : (x - α)^{2} + (y - β)^{2} = r_{1}^{2} and C_{2} : (x - 8)^{2} + (y - \frac{15}{2})^{2} = r_{2}^{2} touch each other externally at the point (6, 6) . If the point (6, 6) divides the line segment joining the centres of the circles C_{1} and C_{2} internally in the ratio 2 : 1, then (α + β) + 4 (r_{1}^{2} + r_{2}^{2}) equals$ (1) $125$ (2) $130$ (3) $110$ (4) $145$ 154. $Let C be a circle passing through the points A(2, -1) and B(3, 4). The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle (x - 5)^{2} + (y - 1)^{2} = \frac{13}{2}, then r^{2} is equal to :$ (1) $32$ (2) $\frac{65}{2}$ (3) $\frac{61}{2}$ (4) $30$ 155. $The centres of those circles which touch the circle, x^{2} + y^{2} - 8 x - 8 y - 4 = 0, externally and also touch the x -axis, lie on :$ (1) $an ellipse which is not a circle$ (2) $a hyperbola$ (3) $a parabola$ (4) $a circle$ 156. $Let T be the line passing through the points P(-2, 7) and Q(2, -5). Let F_{1} be the set of all pairs of circles (S_{1}, S_{2}) such that T is tangent to S_{1} at P and tangent to S_{2} at Q, and also such that S_{1} and S_{2} touch each other at a point, say, M. Let E_{1} be the set representing the locus of M as the pair (S_{1}, S_{2}) varies in F_{1} . Let the set of all straight line segments joining a pair of distinct points of E_{1} and passing through the point R(1, 1) be F_{2} . Let E_{2} be the set of the mid-points of the line segments in the set F_{2} . Then, which of the following statement(s) is (are) TRUE$ (1) $The point (- 2, 7) lies in E_{1}$ (2) $The point (\frac{4}{5}, \frac{7}{5}) does NOT lie in E_{2}$ (3) $The point (\frac{1}{2}, 1) lies in E_{2}$ (4) $The point (0, \frac{3}{2}) does NOT lie in E_{1}$ 157. $For the four circles M, N, O and P, following four equations are given : Circle M : x^{2} + y^{2} = 1 Circle N : x^{2} + y^{2} - 2 x = 0 Circle O : x^{2} + y^{2} - 2 x - 2 y + 1 = 0 Circle P : x^{2} + y^{2} - 2 y = 0 If the centre of circle M is joined with centre of the circle N, further centre of circle N is joined with centre of the circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a :$ (1) $Rhombus$ (2) $Square$ (3) $Rectangle$ (4) $Parallelogram$ 158. $If one of the diameters of the circle, given by the equation, x^{2} + y^{2} - 4 x + 6 y - 12 = 0, is a chord of a circle S, whose centre is at (- 3, 2), then the radius of S is :$ (1) $53$ (2) $5$ (3) $10$ (4) $52$ 159. $Let the tangent to the circle C_{1} : x^{2} + y^{2} = 2 at the point M(-1,1) intersect the circle C_{2} : (x - 3)^{2} + (y - 2)^{2} = 5, at two distinct points A and B. If the tangents to C_{2} at the points A and B intersect at N, then the area of the triangle ANB is equal to :$ (1) $\frac{1}{2}$ (2) $\frac{2}{3}$ (3) $\frac{1}{6}$ (4) $\frac{5}{3}$ 160. $Two tangents are drawn from the point P(-1, 1) to the circle x^{2} + y^{2} - 2x - 6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to:$ (1) $2$ (2) $(32 + 2)$ (3) $4$ (4) $3 (2 - 1)$ 161. $Consider two circles C_{1} : x^{2} + y^{2} = 25 and C_{2} : (x - α)^{2} + y^{2} = 16, where α \in (5, 9) . Let the angle between the two radii (one to each circle) drawn from one of the intersection points of C_{1} and C_{2} be sin^{- 1} (\frac{63}{8}) . If the length of common chord of C_{1} and C_{2} is β, then the value of (αβ)^{2} equals$ 162. $Let C be the circle x^{2} + (y - 1)^{2} = 2, E_{1} and E_{2} be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line x + y = 3 touch the curves C, E_{1} and E_{2} at P (x_{1}, y_{1}), Q (x_{2}, y_{2}) and R (x_{3}, y_{3}) respectively. Given that P is the mid point of the line segment QR and PQ = \frac{2 2}{3}, the value of 9 (x_{1} y_{1} + x_{2} y_{2} + x_{3} y_{3}) is equal to______.$ 163. $Let ABCD be a square of side of unit length.Let a circle C_{1} centered at A with unit radius is drawn. Another circle C_{2} which touches C_{1} and the lines AD and AB are tangent to it, isalso drawn. Let a tangent line from the point C to the circle C_{2} meet the side AB at E . If the length of EB is α + 3 β, where α, β are integers, then α + β is equal to$ 164. $Let AB be a chord of length 12 of the circle (x - 2)^{2} + (y + 1)^{2} = \frac{169}{4} . If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to___.$